Translation (Geometry)
In short: Translation is the motion that carries all points of a shape by one common distance along one common direction, leaving size, form and facing untouched while position shifts.
8th GradeGeometry8.G.A.1
What is Translation?
Picture a chess piece pushed across the board without twisting it: that pure slide is a translation. Every point travels the same distance in the same direction, so the new figure is an exact copy resting somewhere else.
On a coordinate plane, a shift of \(a\) units sideways and \(b\) units vertically sends \((x, y)\) to \((x + a, y + b)\). A positive \(a\) goes right, a negative \(a\) goes left; a positive \(b\) goes up, a negative \(b\) goes down. Lengths and angles never change, so the image is congruent to the original.
How to spot it
- The wording says slide, shift or push a shape by so many units right, left, up or down.
- Coordinates arrive with an arrow rule like (x, y) → (x + 5, y − 3).
- One figure repeats along a row, like tiles on a floor or a decorative border.
- A picture shows an original and its copy facing the same way, merely displaced.
The method
- Turn the instruction into a sideways shift \(a\) and an up-down shift \(b\), signs included.
- Add \(a\) to each x-coordinate and \(b\) to each y-coordinate of the corners.
- Mark the new corners and join them in the original order.
- Verify that every side kept its length and direction, so the copy faces the way the original did.
Translation: solved example
Problem
A sail-shaped triangle ABC has corners \(A(1, 2)\), \(B(4, 2)\) and \(C(1, 5)\). Slide it 5 units right and 3 units down. Where do the corners land?
Solution
The slide is encoded by \((x, y) \to (x + 5, y – 3)\).
Vertex by vertex: \(A^{\prime} = (1 + 5, 2 – 3) = (6, -1)\), \(B^{\prime} = (4 + 5, 2 – 3) = (9, -1)\), \(C^{\prime} = (1 + 5, 5 – 3) = (6, 2)\). So the image has corners \((6, -1)\), \((9, -1)\) and \((6, 2)\), and every side keeps its length from ABC.
Try it
Question 1
Where does \((-3, 4)\) land after a slide of 7 left and 2 up?
Show the answer
\((-3 – 7, 4 + 2) = (-10, 6)\).
Question 2
A point moves from \((2, 9)\) to \((8, 5)\). Write the rule.
Show the answer
The x-value rose by 6 and the y-value fell by 4, so \((x, y) \to (x + 6, y – 4)\).
Translation vs. Reflection
A translation slides a figure and keeps it facing the same way. A reflection flips it across a line, producing a mirror image that faces the opposite way. Both are rigid motions that preserve lengths and angles, but a translation never reverses orientation, and a reflection always does.
Keep going
Lessons
Worksheets and quizzes
- Translations, Reflections and Rotations – Worksheet
- Translations, Reflections and Rotations – Quiz
- Translations, Reflections and Rotations – Worksheet
- Translations, Reflections and Rotations – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Geometry